Abstract
In this article, we review the construction of Hamiltonian perturbation theories with emphasis on Hori’s theory and its extension to the case of dynamical systems with several degrees of freedom and one resonant critical angle. The essential modification is the comparison of the series terms according to the degree of homogeneity in both \(\sqrt \varepsilon \) and a parameter which measures the distance from the exact resonance, instead of just \(\sqrt \varepsilon \).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Bohlin, K.: 1888, “Über eine neue Annäherungsmetode in der Störungstheorie”, Bihang Svenska Vet.-Akad. 14 Afd.I No.5, 1–26.
Brouwer, D.: 1959, “Solution of the problem of artificial satellite theory without drag”, Astron. J. 64, 378–397.
Brouwer, D. and Clemence, G.M.: 1961, Methods of Celestial Mechanics, Academic Press, New York.
Campbell, J.A. and Jefferys, W.H.: 1970, “Equivalence of the perturbation theories of Hori and Deprit”, Celest. Mech. 2, 467–473.
Carathéodory, C.: 1965, Calculus of Variations and Partial Differential Equations of the First Order, Holden-Day, San Francisco.
Deprit. A.: 1969, “Canonical transformation depending on a small parameter”, Celest. Mech. 1, 12–30.
Ferraz-Mello, S.: 1990, “Averaging Hamiltonian systems”, in: Modern Methods in Celestial Mechanics (D. Benest, Cl. Froeschlé, eds), Ed. Frontières, pp.151–211.
Ferraz-Mello, S.:1997, Hamiltonian Averaging Theories and Resonance in Celestial Mechanics, to be published.
Garfinkel, B., Jupp, A. and Williams, C.: 1971, “A recursive Von Zeipel algorithm for the ideal resonance problem”, Astron. J. 76, 157.
Gröbner, W.: 1960, Die Lie Reihen und ihre Anwendungen, Deutsche Verl. Wiss., Berlin.
Hori, G.-I.: 1966, “Theory of general perturbations with unspecified canonical variables”, Publ. Astron. Soc. Japan, 18, 287–296.
Jupp, A.: 1970, “On the ideal resonance problem — II”, Mon. Not. Roy. Astron. Soc. 148, 197–210.
Jupp, A.: 1982, “A comparison of the Brouwer-Von Zeipel and Brouwer-Lie series methods in resonant systems”, Celest. Mech. 26, 413–422.
Jupp, A.:1992, “Chapter XIX — Bohlin Methods”, Not. Scient. Techn. Bur. Longit. No. S 036.
Mersman, W. A.: 1970, “A new algorithm for the Lie transforms” Celest. Mech. 3, 81–89.
Poincaré, H.:1893, Les Méthodes Nouvelles de la Mécanique Celeste, Gauthier-Villars, Paris, Vol. II.
Sérsic, J.L.:1956, Aplicaciones de un Certo Tipo de Transformaciones Canónicas a la Mecánica Celeste, Dr. Thesis, Univ. Nac. La Plata, (repr. Serie Astronomica, Obs. Astron. Univ. La Plata 35, 1969).
Sessin, W.: 1986, “A theory of integration for the extended Delaunay method”, Celest. Mech. 39, 173–180.
Von Zeipel, H.:1916, “Recherches sur le mouvement des petites planètes”, Arkiv. Mat. Astron. Fysik, 11-13.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1997 Springer Science+Business Media Dordrecht
About this paper
Cite this paper
Ferraz-Mello, S. (1997). On Hamiltonian Averaging Theories and Resonance. In: Wytrzyszczak, I.M., Lieske, J.H., Feldman, R.A. (eds) Dynamics and Astrometry of Natural and Artificial Celestial Bodies. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5534-2_5
Download citation
DOI: https://doi.org/10.1007/978-94-011-5534-2_5
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-6330-2
Online ISBN: 978-94-011-5534-2
eBook Packages: Springer Book Archive